xaddge0 - Intuitionistic Logic Explorer

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Theorem xaddge0 9947

Description: The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)

**Assertion**
Ref	Expression
xaddge0	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +_𝑒 𝐵))

**Proof of Theorem xaddge0**
Step	Hyp	Ref	Expression
1		0xr 8068	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 9	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ ℝ^)
3		simplr 528	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ^)
4		xaddcl 9929	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
5	4	adantr 276	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
6		simprr 531	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
7		xaddid2 9932	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
8	3, 7	syl 14	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (0 +_𝑒 𝐵) = 𝐵)
9		simpll 527	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐴 ∈ ℝ^)
10		simprl 529	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐴)
11		xleadd1a 9942	. . . 4 ⊢ (((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 0 ≤ 𝐴) → (0 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
12	2, 9, 3, 10, 11	syl31anc 1252	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (0 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
13	8, 12	eqbrtrrd 4054	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ≤ (𝐴 +_𝑒 𝐵))
14	2, 3, 5, 6, 13	xrletrd 9881	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +_𝑒 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 class class class wbr 4030 (class class class)co 5919 0cc0 7874 ℝ^*cxr 8055 ≤ cle 8057 +_𝑒 cxad 9839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-i2m1 7979 ax-0id 7982 ax-rnegex 7983 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-xadd 9842

This theorem is referenced by: ge0xaddcl 10052

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